Fibration on the category of Lie pseudoalgebras implementing comorphisms

Question

I am trying to understand comorphisms of Lie pseudoalgebras from the point of view of fibred categories, but failing miserably so far. My question would be:

Is there a (op)fibration $\mathrm{LiePs} \to \mathrm{Alg}$ of the category of Lie pseudoalgebras in the sense of (op)fibred categories such that comorphisms of Lie pseudoalgebras are morphisms in the dual fibration $\mathrm{LiePs}^*$?

Let me be a bit more precise: Lie pseudoalgebras (aka Lie-Rinehart algebras or others) can be seen as the algebraic counterparts of Lie algebroids.

Definition: (Lie Pseudoalgebra)
A Lie pseudoalgebra consists of a commutative algebra $\mathcal{A}$ and a Lie algebra $\mathfrak{g}$, such that $\mathfrak{g}$ is an $\mathcal{A}$-module and $\mathcal{A}$ acts on $\mathfrak{g}$ by derivations, i.e. we have a Lie algebra morphism $\rho \colon \mathfrak{g} \to \mathrm{Der}(\mathcal{A})$ which is also an $\mathcal{A}$-module morphism and we have $$ [\xi_1, a \xi_2] = a[\xi_1,\xi_2] + \rho(\xi_1)(a)\xi_2$$ for $\xi_1, \xi_2 \in \mathfrak{g}$, $a \in \mathcal{A}$.

For Lie pseudoalgebras there is a quite obvious notion of morphism.

Definition: (Morphism of Lie pseudoalgebras)
A morphism of Lie pseudoalgebras $(\mathcal{A},\mathfrak{g})$ and $(\mathcal{B},\mathfrak{h})$ consists of an algebra morphism $\phi \colon \mathcal{A} \to \mathcal{B}$ and a module morphism $\Phi \colon \mathfrak{g} \to \mathfrak{h}$ along $\phi$ which is also a Lie algebra morphism and $$ \phi(\rho_{\mathcal{A}}(\xi)a) = \rho_{\mathcal{B}}(\Phi(\xi))\phi(a)$$ holds for $\xi \in \mathfrak{g}$, $a \in \mathcal{A}$.

Let us denote the category of Lie pseudoalgebras by $\mathrm{LiePs}$. Then there is an obvious functor $\mathrm{LiePs} \to \mathrm{Alg}$ by mapping a Lie pseudoalgebra $(\mathcal{A},\mathfrak{g})$ to the algebra $\mathcal{A}$ and a morphism $(\Phi,\phi)$ to the corresponding algebra morphism $\phi$.

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Question 1: Is $\mathrm{LiePs} \to \mathrm{Alg}$ a (op)fibration of categories?

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If so: for (op)fibred categories there is a notion of a dual (op)fibration, see e.g. A.Kock; The dual fibration in elementary terms, whose morphisms can be understood as comorphisms of the original objects. And there is a notion of comorphism of Lie pseudoalgebras, see e.g. Z. Chen,Z.-J. Liu; On (co-)morphisms of Lie pseudoalgebras and groupoids.

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Question 2: Do comorphisms of Lie pseudoalgebras agree with morphisms in the dual fibration $\mathrm{LiePs}^* \to \mathrm{Alg}$?

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Answer 1

Not a direct answer but this is how I would tackle this:

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Background knowledge: Given a category $C$ and a (weak-) functor $F:C^{op}\to \mathrm{Cat}$ one can construct a category $$\int F$$ (also denoted $\int_C F$) called the Grothendieck Construction that comes with a canonical arrow $\pi_f:\int F\to C$ that happens to be a fibration. Inversely: Given a fibration $\pi: E\to C$ we can construct a (weak-) functor $F_\pi:C^{op}\to \mathrm{Cat}$. These two constructions are inverse in a suitable way. Similarly for covariant functors $F:C\to\mathrm{Cat}$.

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Now: To check if your functor $\mathrm{LiePs}\to\mathrm{Alg}$ is a fibration I would try to construct $\mathrm{LiePs}$ as the Grothendieck Construction of a functor $$Ps:\mathrm{Alg}^{op}\to\mathrm{Cat}$$ (or maybe $Ps:\mathrm{Alg}\to\mathrm{Cat}$).

The object part of this functor should be

$$Ps: A \mapsto \mathrm{Lie}^*_{/\mathrm{Der}(A)}$$ where $\mathrm{Lie}^*_{/\mathrm{Der}(A)}$ is a suitable subcategory of the comma category $\mathrm{Lie}_{/\mathrm{Der}(A)}$. A nice feature of the Grothendieck Construction is the following: If your functor actually is a fibration, you should then be able to extend this assignment to a weak functor in a more or less obvious way: For every morphism $f:A\to B$ in $\mathrm{Alg}$ you would search for a functor $f^*:\mathrm{Ps}(B)\to \mathrm{Ps}(B)$ (Or maybe covariantly $f_*:\mathrm{Ps}(A)\to \mathrm{Ps}(B)$?).

The next step is to check if the resulting category $\int Ps$ is the "same" as $\mathrm{LiePs}$. From the construction of $\int Ps$ there should already by suitable candidates for the equivalence functors.